# Riesz–Markov–Kakutani representation theorem for compact non-Hausdorff spaces

Let $$X$$ be a compact Hausdorff topological space, and $$\mathcal C^0 (X) = \{f:X\to\mathbb{R}; \ f \text{ is continuous }\}$$. It is well known that for any bounded linear functional $$\phi: \mathcal C^0(X)\to\mathbb{R},$$ such that $$\phi(f)\geq 0$$ if $$f\geq 0$$ ($$\phi$$ is called a positive linear functional), then there exists a unique regular Borel measure $$\mu$$, such that $$\phi(g) = \int g\ \mathrm d\mu, \ \forall \ g\in \mathcal C^0(X).$$ This result follows from a direct application of Riesz–Markov–Kakutani representation theorem.

If we drop the Hausdorff hypothesis (only assuming $$X$$ as compact topological space). Then we can lose the uniqueness of the measure that represents the linear functional. A famous example is the compact topological space "$$[0,1]$$ with to origins". In this case the functional $$\phi: \mathcal C^0(X)\to\mathbb{R}$$, $$\phi(f) = f(0)$$ can be written as $$\int f\ \mathrm{d}\delta_0$$ or $$\int f\ \mathrm{d}\delta_{0'}.$$

I would like to know if we still have the existence of a measure that represents the functional. In other words, I would like to know if the following theorem is true

Possible Theorem: Let $$(X,\tau)$$ be a compact non-Hausdorff space, and $$\Lambda : \mathcal C^0(X)\to\mathbb{R}$$ a positive bounded linear functional, then there exists a measure $$\mu: \mathcal B(\tau)\to \mathbb{R}$$ (where $$\mathcal B(\tau)$$ is the smallest $$\sigma$$-algebra such that $$\tau\subset \mathcal B(\tau))$$, such that $$\Lambda(f) = \int f\ \mathrm{d}\mu, \ \forall \ f\in \mathcal C^0(X).$$

Can anyone help me?

I have searched online but I was not able to find a result in the non-Hausdorff case.

First, it follows from the following result and the Riesz–Markov–Kakutani representation theorem that we can always find a suitable Baire measure representing a positive linear functional.

Theorem: Let $$X$$ be any topological space. Then there exists a completely regular Hausdorff space $$Y$$ and a continuous surjection $$\tau:X\to Y$$ such that the function $$g\mapsto g\circ\tau$$ is an isomorphism from $$C_B(Y)$$ onto $$C_B(X)$$.

This is Theorem 3.9 of "Rings of continuous functions" (1960) by Gillman and Jerison.

So the problem reduces to the question whether a Baire measure on a compact topological space can be extended to a Borel measure. We can do this using the following result, which specializes the very abstract Theorem 2.6.1 of "Convex Cones" (1981) by Fuchssteiner and Lusky.

Theorem: Let $$X$$ be a non-empty compact topological space and $$L:\mathcal{C}^0_+(X)\to\mathbb{R}$$ be an additive function on the cone of nonnegative continuous functions on $$X$$ such that $$L(g)\leq\max g$$ for all $$g$$. Then there exists a Borel probability measure $$\nu$$ on $$X$$ such that $$L(g)\leq\int g~\mathrm d\nu$$ for all $$g\in \mathcal{C}^0_+(X)$$.

For nonzero $$\Lambda$$, let $$L=1/\Lambda(1)\cdot \Lambda$$. Then the measure $$\mu=\Lambda(1)\cdot\nu$$ does the trick.

It should be noted that the resulting Borel measure need not be regular. For non-Hausdorff $$X$$, there is no point in going beyond Baire measures.

• Just one question that I am not following (btw, the trick of inducing a Baire-measure using $\tau$ was really clever). Using the second theorem we are able to find a Borel probability measure such that $L(f) \leq \int f \ \mathrm{d}\nu$. Why this solve the problem? Since we are interested in the equality of both terms. – Matheus Manzatto May 17 at 18:09
• @MatheusManzatto I wrote the statement with an inequality, so that it matches up easily with the statement in the book. Here is how one can get equality: Assume without loss of generality that $\Lambda(1)=1$ and $0\leq g\leq 1$ (the general case follows from rescaling). Then $\Lambda(g)\leq\int g~\mathrm d\nu$ and $\Lambda(1-g)\leq \int 1-g~\mathrm d\nu$ together with $1=\Lambda(1)=\Lambda(g)+\Lambda(1-g)\leq \int g~\mathrm d\nu+\int 1-g~\mathrm d\nu=1$ implies that $\Lambda(g)=\int g~\mathrm d\nu$. – Michael Greinecker May 17 at 18:19
• Wow, that was good. Thx very much for your help. – Matheus Manzatto May 17 at 18:23
• Nice. It seems to me from you answer and your comments that the Corollary to Theorem II.2.6.1 from the book of Fuchssteiner and Lusky does the job alone - it's even stated in the book as a Riesz representation theorem for non-Hausdorff compact topological spaces. Is the "Tychonoffication" result from Gilman-Jerison even needed? – Pedro Lauridsen Ribeiro May 18 at 6:09
• @PedroLauridsenRibeiro You are right. The first result is for context. – Michael Greinecker May 18 at 6:34